Problem: Determine how many solutions exist for the system of equations. ${-x+y = 5}$ ${y = 7+x}$
Answer: Convert both equations to slope-intercept form: ${-x+y = 5}$ $-x{+x} + y = 5{+x}$ $y = 5+x$ ${y = x+5}$ ${y = 7+x}$ ${y = x+7}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = x+5}$ ${y = x+7}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.